the order of a PDE

If you had to sort PDEs into bins by difficulty and behaviour, the very first label you would reach for is the order. The order of a PDE is simply the order of the highest derivative that appears in it. A first-order equation involves at most first derivatives (u_x, u_t); a second-order equation has a second derivative somewhere (u_xx, u_tt, or a mixed u_xy) but nothing higher. It is the easiest property to read off — you scan the equation and find the most-differentiated term.

Counting is straightforward but watch the mixed terms: u_xy is a second-order term because it involves two differentiations total, even though each is only first-order in its own variable. So u_t + u_x = 0 is first order; u_t = k u_xx is second order; the biharmonic equation Laplacian(Laplacian u) = 0, which works out to fourth derivatives like u_xxxx, is fourth order. The order does not care how many independent variables there are or whether the equation is linear — only how deeply you differentiate.

Order matters because it largely controls how much data a problem needs and how the equation behaves. Roughly, the higher the order, the more conditions you must supply to single out a solution — an ODE of order n needs n constants, and a PDE's data requirements scale with its order too (though for PDEs the 'constants' become whole functions). The vast majority of the equations of classical physics are second order — heat, wave, Laplace, Poisson all are — which is why the deep classification into elliptic, parabolic and hyperbolic types is built around second-order equations. First-order equations form their own world, solved by following characteristic curves.